ACTA ARITHMETICA
Online First version
The set of minimal distances in Krull monoids
by
Alfred Geroldinger and Qinghai Zhong (Graz)
1. Introduction. Let H be a Krull monoid with class group G (we have
in mind holomorphy rings in global fields and give more examples later).
Then every nonunit of H has a factorization as a finite product of atoms
(or irreducible elements), and all these factorizations are unique (i.e., H is
factorial) if and only if G is trivial. Otherwise, there are elements having
factorizations which differ not only up to associates and up to the order of
the factors. These phenomena are described by arithmetical invariants such
as sets of lengths and sets of distances. We first recall some concepts and
then we formulate a main result of the present paper.
For a finite nonempty set L = {m1 , . . . , mk } of positive integers with
m1 < · · · < mk , we denote by ∆(L) = {mi − mi−1 | i ∈ [2, k]} the set of
distances of L. Thus ∆(L) = ∅ if and only if |L| ≤ 1. If a nonunit a ∈ H
has a factorization a = u1 · . . . · uk into atoms u1 , . . . , uk , then k is called
the length of the factorization, and the set LH (a) = L(a) of all possible k is
called the set of lengths of a. If there is an element a ∈ H with |L(a)| > 1,
then it immediately follows that |L(an )| > n for every n ∈ N. Since H is
Krull, every nonunit has a factorization into atoms and all sets of lengths
are finite. The set of distances ∆(H) is the union of all sets ∆(L(a)) over all
nonunits a ∈ H. Thus, by definition, ∆(H) = ∅ if and only if |L(a)| = 1 for
all nonunits a ∈ H, and ∆(H) = {d} if and only if L(a) is an arithmetical
progression with difference d for all nonunits a ∈ H. The set of minimal
distances ∆∗ (H) is defined as
∆∗ (H) = {min ∆(S) | S ⊂ H is a divisor-closed submonoid with ∆(S) 6= ∅}.
By definition, we have ∆∗ (H) ⊂ ∆(H), and ∆∗ (H) = ∅ if and only if
∆(H) = ∅. If the class group G is finite, then ∆(H) is finite and sets of
2010 Mathematics Subject Classification: 11B30, 11R27, 13A05, 20M13.
Key words and phrases: nonunique factorizations, sets of distances, Krull monoids, zerosum sequences, cross numbers.
Received 25 August 2014; revised 25 February 2016.
Published online *.
DOI: 10.4064/aa7906-1-2016
[1]
c Instytut Matematyczny PAN, 2016
2
A. Geroldinger and Q. Zhong
lengths have a well-defined structure which is given in the next theorem [13,
Chapter 4.7].
Theorem A. Let H be a Krull monoid with finite class group. Then
there is a constant M ∈ N such that the set of lengths L(a) of any nonunit
a ∈ H is an AAMP (almost arithmetical multiprogression) with difference
d ∈ ∆∗ (H) and bound M .
The structural description given above is best possible [32]. The set
of minimal distances ∆∗ (H) has been studied by Chapman, Geroldinger,
Halter-Koch, Hamidoune, Plagne, Smith, Schmid and others, and there are
a variety of results. We refer the reader to the monograph [13, Chapter 6.8]
for an overview and mention some results which have appeared since then.
Suppose that G is finite and that every class contains a prime divisor. Then
the set of distances ∆(H) is an interval [18]. A simple example shows that
the interval [1, r(G) − 1] is contained in ∆∗ (H) (Lemma 2.3) and thus, by
Theorem 1.1 below, ∆∗ (H) is an interval too if r(G) ≥ exp(G) − 1. Cyclic
groups stand in sharp contrast to this. Indeed, if G is cyclic with |G| > 3,
then max(∆∗ (H) \ {|G| − 2}) = b|G|/2c − 1 (see [14]). A detailed study of
the structure of ∆∗ (H) for cyclic groups is given in a recent paper by Plagne
and Schmid [23].
The goal of the present paper is to study the maximum of ∆∗ (H), and
here is the main direct result.
Theorem 1.1. Let H be a Krull monoid with class group G.
(1) If |G| ≤ 2, then ∆∗ (H) = ∅.
(2) If 2 < |G| < ∞, then
max ∆∗ (H) ≤ max{exp(G) − 2, r(G) − 1}
where r(G) denotes the rank of G.
(3) Suppose that every class contains a prime divisor. If G is infinite,
then ∆∗ (H) = N, while if 2 < |G| < ∞, then
max ∆∗ (H) = max{exp(G) − 2, r(G) − 1}.
Theorem 1.1 will be complemented by an associated inverse result (Theorem 4.5) describing how max ∆∗ (H) is attained and disproving a former
conjecture (Remark 4.6). Both the direct and the inverse result have numbertheoretic relevance beyond the occurrence in Theorem A. Indeed, they are
key tools in the characterization of those Krull monoids whose systems of
sets of lengths are closed under set addition [17], in the study of arithmetical characterizations of class groups via sets of lengths [13, Chapter 7.3],
[31, 16], as well as in the asymptotic study of counting functions associated
to periods of sets of lengths [30], [13, Theorem 9.4.10].
The set of minimal distances in Krull monoids
3
In Section 2 we gather the required background from the theory of Krull
monoids and from additive combinatorics. In particular, we indicate that
the set of minimal distances of H equals the set of minimal distances of an
associated monoid of zero-sum sequences (Lemma 2.1), and therefore it can
be studied with methods from additive combinatorics. The proof of Theorem
1.1 will be given in Section 3 and the associated inverse result will be given
in Section 4.
2. Background on Krull monoids and on additive combinatorics.
We denote by N the set of positive integers, and, for a, b ∈ Z, we denote by
[a, b] = {x ∈ Z | a ≤ x ≤ b} the discrete, finite interval between a and b. We
use the convention that max ∅ = 0. By a monoid, we mean a commutative
semigroup with identity that satisfies the cancellation laws. If H is a monoid,
then H × denotes the unit group, q(H) the quotient group, and A(H) the
set of atoms (or irreducible elements) of H. A submonoid S ⊂ H is called
divisor-closed if a ∈ S, b ∈ H, and b divides a imply that b ∈ S. A monoid
H is said to be
• atomic if every nonunit can be written as a finite product of atoms;
• factorial if it is atomic and every atom is prime;
• half-factorial if it is atomic and |L(a)| = 1 for each nonunit a ∈ H
(equivalently, ∆(H) = ∅);
• decomposable if there exist submonoids H1 , H2 with Hi 6⊂ H × for
i ∈ [1, 2] such that H = H1 × H2 (and H is called indecomposable
otherwise).
A monoid F is factorial with F × = {1} if and only if it is free abelian. If
this holds, then the set of primes P ⊂ F is a basis of F , we write F = F(P ),
and every a ∈ F has a representation of the form
Y
a=
pvp (a) with vp (a) ∈ N0 and vp (a) = 0 for almost all p ∈ P.
p∈P
A monoid homomorphism θ : H → B is called a transfer homomorphism
if it has the following properties:
(T1) B = θ(H)B × and θ−1 (B × ) = H × .
(T2) If u ∈ H, b, c ∈ B and θ(u) = bc, then there exist v, w ∈ H such
that u = vw, θ(v) ' b and θ(w) ' c.
If H and B are atomic monoids and θ : H → B is a transfer homomorphism,
then (see [13, Chapter 3.2])
LH (a) = LB (θ(a))
for all a ∈ H,
∆(H) = ∆(B),
∆∗ (H) = ∆∗ (B).
Krull monoids. A monoid H is said to be a Krull monoid if it satisfies
the following two conditions:
4
A. Geroldinger and Q. Zhong
(a) There exists a monoid homomorphism ϕ : H → F = F(P ) into a
free abelian monoid F such that a | b in H if and only if ϕ(a) | ϕ(b)
in F .
(b) For every p ∈ P , there exists a finite subset E ⊂ H such that
p = gcd(ϕ(E)).
Let H be a Krull monoid and ϕ : H → F(P ) a homomorphism satisfying
properties (a) and (b). Then ϕ is called a divisor theory of H,
G = q(F )/q(ϕ(H))
is the class group, and
GP = {[p] = pq(ϕ(H)) | p ∈ P } ⊂ G
the set of classes containing prime divisors. The class group will be written
additively, and the tuple (G, GP ) is uniquely determined by H. To provide
some examples of Krull monoids, we recall that a domain is a Krull domain if
and only if its multiplicative monoid of nonzero elements is a Krull monoid,
and that a noetherian domain is Krull if and only if it is integrally closed.
Rings of integers, holomorphy rings in algebraic function fields, and regular
congruence monoids in these domains are Krull monoids with finite class
group such that every class contains a prime divisor [12], [13, Chapter 2.11].
For monoids of modules and monoid domains which are Krull we refer to
[22, 4, 3, 1].
Next we introduce Krull monoids having a combinatorial flavor which
are used to model arbitrary Krull monoids. Let G be an additively written
abelian group and G0 ⊂ G a subset. An element S = g1 · . . . · gl ∈ F(G0 )
is called a sequence over G0 , σ(S) = g1 + · · · + gl is called its sum, |S| = l
its length, and h(S) = max{vg (S) | g ∈ supp(S)} the maximal multiplicity
of S. The monoid
B(G0 ) = {S ∈ F(G0 ) | σ(S) = 0}
is a Krull monoid, called the monoid of zero-sum sequences over G0 . Its
significance for the study of general Krull monoids is summarized in the
following lemma (see [13, Theorem 3.4.10 and Proposition 4.3.13]).
Lemma 2.1. Let H be a Krull monoid, ϕ : H → D = F(P ) a divisor
theory with class group G, and GP ⊂ G the set of classes containing prime
e : D → F(GP ) denote the unique homomorphism defined by
divisors. Let β
e
e ◦ ϕ : H → B(GP )
β(p)
= [p] for all p ∈ P . Then the homomorphism β = β
is a transfer homomorphism. In particular,
∆∗ (H) = ∆∗ B(GP ) = {min ∆(B(G0 )) | G0 ⊂ GP is a subset such that
B(G0 ) is not half-factorial}.
The set of minimal distances in Krull monoids
5
Thus ∆∗ (H) can be studied in an associated monoid of zero-sum sequences and can thus be tackled by methods of additive combinatorics.
Such transfer results to monoids of zero-sum sequences are not restricted
to Krull monoids, but they also exist for certain seminormal weakly Krull
monoids and certain maximal orders in central simple algebras over global
fields. We do not dwell on this here but refer to [33, Theorem 1.1], [15], and
[2, Section 7].
Zero-sum theory is a subfield of additive combinatorics (see the monograph [20], the survey [10], and for a sample of recent papers on direct
and inverse zero-sum problems with a strong number-theoretic flavor see
[19, 8, 21, 34, 9]). We gather together the concepts needed in what follows.
Let G be a finite abelian group and G0 ⊂ G a subset. Then hG0 i ⊂ G denotes the subgroup generated by G0 . A family (ei )i∈I of elements of G is said
to be independent if ei 6= 0 for all i ∈ I and, for every family (mi )i∈I ∈ Z(I) ,
X
mi ei = 0 implies mi ei = 0 for all i ∈ I.
i∈I
L
A family (ei )i∈I is called a basis for G if ei 6= 0 for all i ∈ I and G = i∈I hei i.
The set G0 is said to be independent if the tuple (g)g∈G0 is independent. If
for a prime p ∈ P, rp (G) is the p-rank of G, then
r(G) = max{rp (G) | p ∈ P} is the rank of G, and
X
r∗ (G) =
rp (G) is the total rank of G.
p∈P
The monoid B(G0 ) of zero-sum sequences over G0 is a finitely generated
Krull monoid. It is traditional to set
A(G0 ) := A(B(G0 )),
∆(G0 ) := ∆(B(G0 )),
∆∗ (G0 ) := ∆∗ (B(G0 )).
Clearly, the atoms of B(G0 ) are precisely the minimal zero-sum sequences
over G0 . The set A(G0 ) is finite, and D(G0 ) = max{|S| | S ∈ A(G0 )} is the
Davenport constant of G0 . The set G0 is called
• half-factorial if the monoid B(G0 ) is half-factorial (equivalently, ∆(G0 )
= ∅);
• non-half-factorial if the monoid B(G0 ) is not half-factorial (equivalently, ∆(G0 ) 6= ∅);
• minimal non-half-factorial if ∆(G0 ) 6= ∅ but every proper subset is
half-factorial;
• (in)decomposable if the monoid B(G0 ) is (in)decomposable.
If G0 is not half-factorial, then min ∆(G0 ) = gcd ∆(G0 ) [13, Proposition
1.4.4]. (Maximal) half-factorial and (minimal) non-half-factorial subsets have
found a lot of attention in the literature (see [11, 28, 24, 25, 29, 5, 6]),
6
A. Geroldinger and Q. Zhong
and cross numbers are a crucial tool for their study. For a sequence S =
g1 · . . . · gl ∈ F(G0 ), we call
k(S) =
l
X
i=1
1
∈ Q≥0
ord(gi )
the cross number of S, and
K(G0 ) = max{k(S) | S ∈ A(G0 )}
the cross number of G0 .
The following simple result [13, Proposition 6.7.3] will be used throughout
the paper without further mention.
Lemma 2.2. Let G be a finite abelian group and G0 ⊂ G a subset. Then
the following statements are equivalent:
(a) G0 is half-factorial.
(b) k(U ) = 1 for every U ∈ A(G0 ).
(c) L(B) = {k(B)} for every B ∈ B(G0 ).
In the remainder of this section we gather some simple, partly well-known
results on the set of minimal distances. A proof of the next lemma can be
found in [13, Chapter 6.8], but for better readability we provide the short
argument here.
Lemma 2.3. Let G be a finite abelian group with |G| > 2.
(1) If g ∈ G with ord(g) > 2, then ord(g) − 2 ∈ ∆∗ (G). In particular,
exp(G) − 2 ∈ ∆∗ (G).
(2) If r(G) ≥ 2, then [1, r(G) − 1] ⊂ ∆∗ (G).
(3) Let G0 ⊂ G be a subset.
(a) If there exists U ∈ A(G0 ) with k(U ) < 1, then min ∆(G0 ) ≤
exp(G) − 2.
(b) If k(U ) ≥ 1 for all U ∈ A(G0 ), then min ∆(G0 ) ≤ |G0 | − 2.
Proof. (1) Let g ∈ G with ord(g) = n > 2 and set G0 = {g, −g}. Then
A(G0 ) = {g n , (−g)n , (−g)g}, ∆(G0 ) = {n − 2}, and hence min ∆(G0 ) =
n − 2.
(2) Let s ∈ [2, r(G)]. Then there is a prime p ∈ P such that Cps is isomorphic to a subgroup of G, and it suffices to show that s − 1 ∈ ∆∗ (Cps ). Let
(e1 , . . . , es ) be a basis of Cps and set e0 = e1 + · · · + es and G0 = {e0 , . . . , es }.
Then a simple calculation (details can be found in [13, Proposition 6.8.1])
shows that ∆(G0 ) = {s − 1} and hence min ∆(G0 ) = s − 1.
(3)(a) Let U = g1 · . . . · gl ∈ A(G0 ) with k(U ) < 1 and n = exp(G) (note
ord(gi )
that k(U ) < 1 implies U 6= 0, l ≥ 2 and k(U ) > 1/n). Then Ui = gi
∈
A(G0 ) for all i ∈ [1, l], and
Un =
l
Y
i=1
n/ord(gi )
Ui
The set of minimal distances in Krull monoids
7
Pl
n
implies that nk(U ) =
i=1 n/ord(gi ) ∈ L(U ). Since k(U ) < 1, we have
nk(U ) ∈ [2, n − 1] and min ∆(G0 ) ≤ n − nk(U ) ∈ [1, n − 2].
(3)(b) The proof is similar to that of (3)(a)—see [13, Lemma 6.8.6] for
details.
Lemma 2.3(3) motivates the following definitions (see [30, 31]). A subset
G0 ⊂ G is called an LCN-set (large cross number set) if k(U ) ≥ 1 for each
U ∈ A(G0 ) and
m(G) = max{min ∆(G0 ) | G0 ⊂ G is a non-half-factorial LCN-set}.
Clearly, if G has a non-half-factorial LCN-set, then |G| ≥ 4. The following
result (due to Schmid [31]) is crucial for our approach.
Proposition 2.4. Let G be a finite abelian group with |G| > 2. Then
max ∆∗ (G) = max{exp(G)−2, m(G)},
m(G) ≤ max{r∗ (G)−1, K(G)−1}.
If G is a p-group, then m(G) = r(G) − 1, and thus
max ∆∗ (G) = max{exp(G) − 2, r(G) − 1}.
Proof. See [31, Theorem 3.1, Lemma 3.3(4), and Proposition 3.6].
Lemma 2.5. Let G be a finite abelian group and G0 ⊂ G a subset.
(1) The following statements are equivalent :
(a) G0 is decomposable.
(b) There are nonempty subsets G1 , G2 ⊂ G0 such that G0 = G1 ]G2
and B(G0 ) = B(G1 )×B(G2 ).
(c) There are nonempty G1 , G2 ⊂ G0 such that G0 = G1 ] G2 and
A(G0 ) = A(G1 ) ] A(G2 ).
(d) There are nonempty G1 , G2 ⊂ G0 such that hG0 i = hG1 i ⊕ hG2 i.
(2) If G0 is minimal non-half-factorial, then G0 is indecomposable.
Proof. (1) See [26, Lemma 3.7] and [1, Lemma 3.2].
(2) This follows immediately from (1)(b).
We will use the following simple fact throughout. For every subset G0
in G and every g ∈ G0 , we have
(2.1)
gcd({vg (B) | B ∈ B(G0 )}) = gcd({vg (A) | A ∈ A(G0 )})
= min({vg (A) | vg (A) > 0, A ∈ A(G0 )})
= min({vg (B) | vg (B) > 0, B ∈ B(G0 )})
= min({k ∈ N | kg ∈ hG0 \ {g}i}) = gcd({k ∈ N | kg ∈ hG0 \ {g}i}).
In particular, min({k ∈ N | kg ∈ hG0 \ {g}i}) divides ord(g).
8
A. Geroldinger and Q. Zhong
Lemma 2.6. Let G be a finite abelian group and G0 ⊂ G a subset.
(1) Suppose that for any distinct h, h0 ∈ G0 we have h 6∈ hG0 \ {h, h0 }i.
Then for any atom A with supp(A) ( G0 and any h ∈ supp(A), we
have gcd(vh (A), ord(h)) > 1.
(2) If G0 is minimal non-half-factorial, then there exists a minimal nonhalf-factorial subset G∗0 ⊂ G with |G0 | = |G∗0 | and a transfer homomorphism θ : B(G0 ) → B(G∗0 ) such that:
(a) For each g ∈ G∗0 , we have g ∈ hG∗0 \ {g}i.
(b) For each B ∈ B(G0 ), we have k(B) = k(θ(B)).
(c) If G∗0 has the property that for each h ∈ G∗0 , h 6∈ hEi for any
E ( G∗0 \ {h}, then G0 also has this property.
(d) If G∗0 has the property that there exists h ∈ G∗0 such that G∗0 \{h}
is independent, then G0 also has this property.
Proof. (1) Assume to the contrary that there are A and h as above
such that gcd(vh (A), ord(h)) = 1. Choose h0 ∈ G0 \ supp(A); then h ∈
hsupp(A) \ {h}i ⊂ hG0 \ {h, h0 }i, a contradiction.
(2) By [13, Theorem 6.7.11], there are a subset G∗0 ⊂ G satisfying property (a) and a transfer homomorphism θ : B(G0 ) → B(G∗0 ). Moreover, θ is a
composition of transfer homomorphisms θ0 of the following form:
• Let g ∈ G0 , m = min{k ∈ N | kg ∈ hG0 \{g}i}, G00 = G0 \({g}∪{mg}),
and define
θ0 : B(G0 ) → B(G00 ),
θ0 (B) = g −vg (B) (mg)vg (B)/m B,
It is shown in [13] that m | vg (B) and m | ord(g).
Therefore it is sufficient to show that |G0 | = |G00 | and that θ0 satisfies properties (b)–(d).
(i) By definition, we have k(B) = k(θ0 (B)) for all B ∈ B(G0 ).
(ii) Since G0 is a minimal non-half-factorial set, the same is true for G00
by [13, Lemma 6.8.9]. If mg ∈ G0 \ {g}, then G00 ( G0 would be non-halffactorial, contrary to the minimality of G0 . Hence mg 6∈ G0 \ {g}, which
implies that |G00 | = |G0 |.
(iii) We set G0 = {g = g1 , . . . , gk } (note that k ≥ 2), G00 = {mg, g2 , . . . , gk },
and suppose that h 6∈ hEi for each h ∈ G00 and for any E ( G00 \{h}. Assume
to the contrary that there exist h ∈ G0 and E ( G0 \ {h} such that h ∈ hEi.
If h = g, then mg ∈ hEi, a contradiction.
Suppose that h 6= g, say h = gk ∈ hEi with E ( {g, g2 , . . . , gk−1 }. If
g 6∈ E, then E ( G00 \ {mg}, a contradiction. Thus
P g ∈ E, and we set
0
E = E \ {g} ∪ {mg}. Since h ∈ hEi, we have h = x∈E\{g} tx x + tg where
P
tx , t ∈ Z. Thus tg = h − x∈E\{g} tx x ∈ hE ∪ {h} \ {g}i ⊂ hG0 \ {g}i.
The set of minimal distances in Krull monoids
9
P
By (2.1), we deduce that m | t and hence h = x∈E\{g} tx x + mt mg ∈ hE 0 i,
a contradiction.
(iv) We set G0 = {g = g1 , . . . , gk }, G00 = {mg, g2 , . . . , gk }, and suppose
that there exists h ∈ G00 such that G00 \ {h} is independent. If h = mg, then
G0 \ {g} = G00 \ {h} is independent. Suppose that h 6= mg, say h = gk . Then
{mg, g2 , . . . , gk−1 } is independent and assume to the contrary that G0 \ {h}
= {g, g2 , . . . , gk−1 } is not independent. Then there exist t1 , . . . , tk−1 ∈ Z
such that t1 g + t2 g2 + · · · + tk−1 gk−1 = 0 but ti gi 6= 0 for at least one
i ∈ [1, k − 1]. This implies that t1 g ∈ hg2 , . . . , gk−1 i ⊂ hG0 \ {g}i. By (1),
t1
we infer that m | t1 and hence m
mg + t2 g2 + · · · + tk−1 gk−1 = 0, contrary to
{mg, g2 , . . . , gk−1 } being independent.
3. Direct results on ∆∗ (H)
Lemma 3.1. Let G be a finite abelian group and G0 ⊂ G a subset with
|G0 | ≥ r(G) + 2 such that:
(a) For each h ∈ G0 , G0 \ {h} is half-factorial and h 6∈ hG0 \ {h, h0 }i for
any h0 ∈ G0 \ {h}.
(b) There exists g ∈ G0 such that g ∈ hG0 \ {g}i and ord(g) is not a
prime power.
Then |G0 | ≤ exp(G) − 2.
Proof. We set exp(G) = n = pk11 ·. . .·pkt t , where t ≥ 2, k1 , . . . , kt ∈ N and
p1 , . . . , pt are distinct primes. By Lemma 2.6(1), we know that for any atom
A with supp(A) ( G0 and any h ∈ supp(A), we have gcd(vh (A), ord(h)) > 1.
In particular,
(3.1)
vh (A) ≥ 2
for each h ∈ supp(A).
We assert the following:
A. For each ν ∈ [1, t] with pν | ord(g), there is an atom Uν ∈ A(G0 ) such
that vg (Uν ) | n/pkνν , k(Uν ) = 1, supp(Uν ) ( G0 , and
|supp(Uν ) \ {g}| ≤ (n − vg (Uν ))/2.
Proof of A. Let ν ∈ [1, t] with pν | ord(g). Since g ∈ hG0 \ {g}i and
t ≥ 2, it follows that 0 6= (n/pkνν )g ∈ Gν = h(n/pkνν )h | h ∈ G0 \ {g}i.
Obviously, Gν is a pν -group. Let Eν ⊂ G0 \ {g} be minimal such that
(n/pkνν )g ∈ h(n/pkνν )Eν i. The minimality implies that |Eν | = |(n/pkνν )Eν |
and (n/pkνν )Eν is a minimal generating set of G0ν := h(n/pkνν )Eν i. Thus [13,
Lemma A.6.2] implies that |(n/pkνν )Eν | ≤ r∗ (G0ν ). Putting all together we
obtain
n
|Eν | = kν Eν ≤ r∗ (G0ν ) = r(G0ν ) ≤ r(G).
pν
10
A. Geroldinger and Q. Zhong
Let dν ∈ N be minimal such that dν g ∈ hEν i. By (2.1), dν | n/pkνν and there
exists an atom Uν such that vg (Uν ) = dν and |supp(Uν )| ≤ |Eν | + 1 ≤
r(G) + 1 ≤ |G0 | − 1, whence supp(Uν ) ( G0 . Thus property (a) implies that
k(Uν ) = 1. Let
Y
Uν = g vg (Uν )
hvh (Uν ) .
h∈supp(Uν )\{g}
Since vh (Uν ) ≥ 2 for each h ∈ supp(Uν ) \ {g} by (3.1), it follows that
vg (Uν )
2
+ |supp(Uν ) \ {g}| ,
1 = k(Uν ) ≥
n
n
whence |supp(Uν ) \ {g}| ≤ (n − vg (Uν ))/2. Proof of A
Let s ∈ N be minimal such that there exists a nonempty subset E (
G0 \ {g} with sg ∈ hEi; suppose E is a minimal such set. By (2.1), there is
an atom V with vg (V ) = s and supp(V ) = {g} ∪ E ( G0 . Then
X vh (V )
s
1 = k(V ) =
+
.
ord(g)
ord(h)
h∈E
By (3.1), we have vh (V ) ≥ 2 for each h ∈ E, and hence the equation above
implies that |E| ≤ (n − s)/2.
Case 1: s is a power of a prime, say a power of p1 . Let E1 = supp(U1 )\{g}.
Since vg (U1 ) | n/pk11 , we have g ∈ hsg, vg (U1 )gi ⊂ hE ∪ E1 i. Property (a)
implies that E ∪ E1 = G0 \ {g}, and thus
vg (U1 ) + s
n − s n − vg (U1 )
|G0 | ≤ 1 + |E| + |E1 | ≤ 1 +
+
=1+n−
.
2
2
2
Since gcd(vg (U1 ), s) = 1, it follows that vg (U1 )+s ≥ 5, hence |G0 | ≤ n−3/2,
and thus |G0 | ≤ n − 2.
Case 2: s is not a prime power, say p1 p2 | s. Then s ≥ 6. Let d =
gcd(s, vg (U1 )) and E1 = supp(U1 )\{g}. Then d < s and dg ∈ hsg, vg (U1 )gi ⊂
hE ∪ E1 i ⊂ hG0 \ {g}i. The minimality of s implies that E ∪ E1 = G0 \ {g},
and thus
n − s n − vg (U1 )
+
|G0 | ≤ 1 + |E| + |E1 | ≤ 1 +
2
2
vg (U1 ) + s
=1+n−
≤ n − 3.
2
Lemma 3.2. Let G be a finite abelian group with exp(G) = n. Let
G0 ⊂ G be a minimal non-half-factorial LCN-set, and suppose that there
is a subset G2 ⊂ G0 such that hG2 i = hG0 i and |G2 | ≤ |G0 | − 2. Then
min ∆(G0 ) ≤ max{1, n − 4}.
Proof. Assume to the contrary that min ∆(G0 ) ≥ max{2, n − 3}. By [27,
Corollary 3.1], the existence of G2 implies that k(U ) ∈ N for each U ∈ A(G0 )
The set of minimal distances in Krull monoids
11
and
min ∆(G0 ) | gcd({k(A) − 1 | A ∈ A(G0 )}).
We set
W1 = {A ∈ A(G0 ) | k(A) = 1},
W2 = {A ∈ A(G0 ) | k(A) > 1}.
Then, for any U1 , U2 ∈ W2 ,
(3.2)
k(U1 ) ≥ max{3, n − 2} and
either k(U1 ) = k(U2 ) or |k(U1 ) − k(U2 )| ≥ max{2, n − 3}.
We choose U ∈ W2 . Then supp(U ) = G0 , and we pick g ∈ G0 \ G2 .
Then g ∈ hG2 i and, by (2.1), there is an atom A with vg (A) = 1 and
supp(A) ⊂ G2 ∪ {g} ( G0 . This implies that A ∈ W1 , and
U Aord(g)−vg (U ) = g ord(g) S
for some zero-sum sequence S over G. Since supp(S) = G0 \ {g} and G0
is minimal non-half-factorial, S has a factorization into a product of atoms
from W1 . Therefore, for each U ∈ W2 , there are A1 , . . . , Am ∈ W1 , where
m ≤ ord(g) − vg (U ) ≤ n − 1, such that U A1 · . . . · Am can be factorized into
a product of atoms from W1 .
We set
W0 = A ∈ A(G0 ) k(A) = min{k(B) | B ∈ W2 } ⊂ W2 ,
and we consider all tuples (U, A1 , . . . , Am ), where U ∈ W0 and A1 , . . . , Am
∈ W1 , such that U A1 · . . . · Am can be factorized into a product of atoms
from W1 . We fix one such tuple (U, A1 , . . . , Am ) with m minimal possible.
Note that m ≤ n − 1. Let
(3.3)
U A1 · . . . · Am = V1 · . . . · Vt
with t ∈ N and V1 , . . . , Vt ∈ W1 .
We observe that k(U ) = t − m and assert the following:
A1. For each ν ∈ [1, t], we have Vν - U A1 · . . . · Am−1 .
Proof of A1. Assume to the contrary that there is a ν ∈ [1, t], say ν = 1,
with V1 | U A1 · . . . · Am−1 . Then there are l ∈ N and T1 , . . . , Tl ∈ A(G0 ) such
that
U A1 · . . . · Am−1 = V1 T1 · . . . · Tl .
By the minimality of m, there exists ν ∈ [1, l] such that Tν ∈ W2 , say ν = 1.
Since
l
X
k(Tν ) = k(U ) + (m − 1) − 1 − k(T1 ) ≤ m − 2 ≤ n − 3,
ν=2
and k(T 0 ) ≥ n − 2 for all T 0 ∈ W2 , it follows that T2 , . . . , Tl ∈ W1 , whence
P
l = 1 + lν=2 k(Tν ) ≤ m − 1. We obtain
V1 T1 · . . . · Tl Am = U A1 · . . . · Am = V1 · . . . · Vt ,
12
A. Geroldinger and Q. Zhong
and thus
T1 · . . . · Tl Am = V2 · . . . · Vt .
The minimality of m implies that k(T1 ) > k(U ). It follows that
k(T1 ) − k(U ) = m − 1 − l ≤ m − 2 ≤ n − 3 ≤ max{n − 3, 2} ≤ k(T1 ) − k(U ).
Therefore l = 1, m = n − 1, n ≥ 5, and k(T1 ) = k(U ) + n − 3. Thus
T1 An−1 = V2 · . . . · Vt ,
and hence
t − 1 ≤ |An−1 |.
This equation shows that k(T1 ) = t − 2 ≤ |An−1 | − 1 ≤ n − 1, and hence
n − 2 ≤ k(U ) = k(T1 ) − n + 3 ≤ 2, contradicting n ≥ 5. Proof of A1
Since exp(G) = n and k(Am ) = 1, it follows that |Am | ≤ n. By A1, for
each ν ∈ [1, t] there exists hν ∈ supp(Am ) such that
vhν (Vν ) > vhν (U A1 · . . . · Am−1 ).
For each h ∈ supp(Am ) we define
Fh = {ν ∈ [1, t] | vh (Vν ) > vh (U A1 · . . . · Am−1 )} ⊂ [1, t].
Thus
[
Fh = [1, t],
h∈supp(Am )
and for each h ∈ supp(Am ), we have
vh (Am ) + vh (U A1 · . . . · Am−1 ) =
t
X
i=1
X
vh (Vi ) ≥
vh (Vi )
i∈Fh
≥ |Fh |(vh (U A1 · . . . · Am−1 ) + 1).
Since |Am | > |supp(Am )| (otherwise Am | U , a contradiction), we obtain
X
[
t=
Fh ≤
|Fh |
h∈supp(Am )
X
≤
h∈supp(Am )
X
≤
h∈supp(Am )
h∈supp(Am )
vh (Am ) + vh (U A1 · . . . · Am−1 )
vh (U A1 · . . . · Am−1 ) + 1
vh (Am ) + 1
|Am | |supp(Am )|
=
+
< |Am | ≤ n.
2
2
2
By (3.3) and (3.2), we have max{3, n − 2} ≤ k(U ) = t − m ≤ n − 1 − m
and hence m = 1, n ≥ 5, t = n − 1, and k(U ) = n − 2. Therefore
(3.4)
U A1 = V1 · . . . · Vn−1 ,
|A1 | = n,
n − 2 ≤ |supp(A1 )| ≤ n − 1,
and
(3.5)
X
h∈supp(A1 )
|Fh | = n − 1,
with Fh , h ∈ supp(A1 ), pairwise disjoint.
The set of minimal distances in Krull monoids
Furthermore, |Fh | ≤
supp(A1 ),
(3.6)
|Fh | ≤ 1
vh (A1 )+vh (U )
vh (U )+1
13
for all h ∈ supp(A1 ). Then for h ∈
when vh (A1 ) ≤ 2,
|Fh | ≤ 2
when vh (A1 ) ≤ 4.
Now we consider all atoms A1 ∈ W1 such that U A1 can be factorized
into a product of n − 1 atoms from W1 , and among them we consider the
atoms A01 for which |supp(A01 )| is minimal; from these we choose an atom
A001 for which h(A001 ) is minimal. Changing notation if necessary we suppose
that A1 has this property. By (3.4), we distinguish three cases depending on
|supp(A1 )| and h(A1 ).
Case 1: |supp(A1 )| = n − 1. Let supp(A1 ) = {g1 , . . . , gn−1 } and A1 =
g12 g2 · . . . · gn−1 . Since h(A1 ) = 2, (3.6) and (3.5) imply that |Fh | = 1 for each
h ∈ supp(A1 ). Note that U g12 g2 · . . . · gn−1 = V1 · . . . · Vn−1 . After renumbering
if necessary we may suppose that Fgi = {i} for each i ∈ [1, n − 1]. Therefore
vgi (Vi ) > vgi (U ) ≥ 1 for each i ∈ [1, n − 1]. Hence vg1 (V1 ) ≥ 2 and we set
V1 = g12 Y1 for some Y1 dividing U . Thus U Y1−1 g2 · . . . · gn−1 = V2 · . . . · Vn−1 ,
which implies that Vi = gi Yi for i ∈ [2, n − 1], where Y2 · . . . · Yn−1 = U Y1−1 .
Summing up we have
(3.7)
U = Y1 · . . . · Yn−1 with Vi = gi Yi for i ∈ [2, n − 1] and V1 = g12 Y1 .
b(n+1)/2c
Since k(A1
g1−n ) = b(n + 1)/2ck(A1 ) − 1 = b(n + 1)/2c − 1 <
b(n+1)/2c −n
max{3, n−2}, it follows that every atom X dividing A1
g1 has cross
b(n+1)/2c −n
number k(X) = 1 by (3.2). Since vg1 (A1
g1 ) ≤ 1, there is an atom C
b(n+1)/2c −n
dividing A1
g1 with supp(C) ⊂ {g2 , . . . , gn−1 } and |supp(C)| ≥ 2,
say g2 , g3 ∈ supp(C). Therefore, V2 V3 = g2 g3 Y2 Y3 | U C, say U C = V2 V3 V 0
for some V 0 ∈ B(G). Since
k(U C) = k(U ) + k(C) = n − 1 = k(V2 ) + k(V3 ) + k(V 0 ),
we obtain k(V 0 ) = n − 3. Now (3.2) implies that V 0 is a product of atoms
from W1 , and hence U C can be factorized into a product of n − 1 atoms.
Since |supp(C)| < n − 1 = |supp(A1 )|, this contradicts the choice of A1 .
Case 2: |supp(A1 )| = n − 2 and h(A1 ) = 2. Let supp(A1 ) = {g1 , . . . , gn−2 }
and A1 = g12 g22 g3 · . . . · gn−2
. Since h(A1 ) = 2, (3.6) implies that |Fh | ≤ 1 for
P
each h ∈ supp(A1 ). Thus h∈supp(A1 ) |Fh | ≤ n − 2, contrary to (3.5).
Case 3: |supp(A1 )| = n − 2 and h(A1 ) = 3. Let supp(A1 ) = {g1 , . . . , gn−2 }
and A1 = g13 g2 ·. . .·gn−2 . Since h(A1 ) = 3, (3.6) and (3.5) imply that |Fg1 | = 2
and |Fgi | = 1 for each i ∈ [2, n−2]. Note that U g13 g2 ·. . .·gn−2 = V1 ·. . .·Vn−1 .
After renumbering if necessary we may suppose that Fg1 = {1, n − 1} and
Fgi = {i} for each i ∈ [2, n − 2]. Therefore vgi (Vi ) > vgi (U ) ≥ 1 for each
i ∈ [1, n − 2] and vg1 (Vn−1 ) > vg1 (U ) ≥ 1. Hence we may set Vn−1 = g12 Yn−1
−1
for some Yn−1 dividing U . Thus U Yn−1
g1 g2 · . . . · gn−2 = V1 · . . . · Vn−2 , which
14
A. Geroldinger and Q. Zhong
−1
implies that Vi = gi Yi for each i ∈ [1, n − 2] where Y1 · . . . · Yn−2 = U Yn−1
.
Summing up we have
(3.8)
U = Y1 ·. . .·Yn−1 with Vi = gi Yi for i ∈ [1, n−2] and Vn−1 = g12 Yn−1 .
b(n+2)/3c
Since k(A1
g1−n ) = b(n + 2)/3ck(A1 ) − 1 = b(n + 2)/3c − 1 <
b(n+2)/3c −n
max{3, n − 2}, it follows that every atom X dividing A1
g1 has
b(n+2)/3c −n
k(X) = 1 by (3.2). Let C ∈ A(G) divide A1
g1 . Then k(C) = 1,
supp(C) ⊂ {g1 , . . . , gn−2 }, and |supp(C)| ≥ 2, say gi , gj ∈ supp(C) where
1 ≤ i < j ≤ n − 2. Therefore Vi Vj = gi gj Yi Yj | U C by (3.8). Arguing as in
Case 1 we infer that U C is a product of n − 1 atoms from W1 . By the choice
of A1 , we obtain |supp(C)| = n − 2 and h(C) ≥ 3. This holds for all atoms
b(n+2)/3c
b(n+2)/3c −n
.
dividing A1
g1 , contradicting the structure of A1
Proof of Theorem 1.1. Let H be a Krull monoid with class group G,
and let GP ⊂ G denote the set of classes containing prime divisors. If
|G| ≤ 2, then H is half-factorial by [13, Corollary 3.4.12], and thus ∆∗ (H) ⊂
∆(H) = ∅. If G is infinite and GP = G, then ∆∗ (H) = N by [7, Theorem
1.1].
Suppose that 2 < |G| < ∞. By Lemma 2.1, it suffices to prove the
statements for the Krull monoid B(GP ). If G is finite, then ∆(G) is finite
by [13, Corollary 3.4.13], hence ∆∗ (G) is finite, and Lemma 2.3 shows that
{exp(G) − 2, r(G) − 1} ⊂ ∆∗ (G).
Since ∆∗ (GP ) ⊂ ∆∗ (G), it remains to prove that
max ∆∗ (G) ≤ max{exp(G) − 2, r(G) − 1}.
Let G0 ⊂ G be a non-half-factorial subset, n = exp(G), and r = r(G). We
need to prove that min ∆(G0 ) ≤ max{n − 2, r − 1}. If G1 ⊂ G0 is non-halffactorial, then min ∆(G0 ) = gcd ∆(G0 ) | gcd ∆(G1 ) = min ∆(G1 ). Thus we
may suppose that G0 is minimal non-half-factorial. If there is an U ∈ A(G0 )
with k(U ) < 1, then Lemma 2.3.3 implies that min ∆(G0 ) ≤ n − 2.
Suppose that k(U ) ≥ 1 for all U ∈ A(G0 ), i.e, G0 is an LCN-set. Since
G0 is minimal non-half-factorial, it follows that G0 is indecomposable by
Lemma 2.5. By Lemma 2.6(2), we may suppose that g ∈ hG0 \ {g}i for all
g ∈ G0 .
Suppose that the order of each element of G0 is a prime power. Since G0
is indecomposable, Lemma 2.5 implies that each order is a power of a fixed
prime p ∈ P, and thus hG0 i is a p-group. By Proposition 2.4 we infer that
min ∆(G0 ) ≤ max ∆∗ (hG0 i) = max{exp(hG0 i) − 2, r(hG0 i) − 1}
≤ max{n − 2, r − 1}.
From now on we suppose that there is a g ∈ G0 whose order is not a
prime power. Then n ≥ 6. If |G0 | ≤ r + 1, then min ∆(G0 ) ≤ |G0 | − 2 ≤ r − 1
The set of minimal distances in Krull monoids
15
by Lemma 2.3.3. Thus we may suppose that |G0 | ≥ r + 2 and we distinguish
two cases.
Case 1: There exists a subset G2 ⊂ G0 such that hG2 i = hG0 i and
|G2 | ≤ |G0 | − 2. Then Lemma 3.2 implies that min ∆(G0 ) ≤ n − 4 ≤ n − 2.
Case 2: Every subset G1 ⊂ G0 with |G1 | = |G0 | − 1 is a minimal generating set of hG0 i. Then for each h ∈ G0 , G0 \ {h} is half-factorial and
h ∈
/ hG0 \ {h, h0 }i for any h0 ∈ G0 \ {h}. Thus Lemma 3.1 implies that
|G0 | ≤ n − 2, and hence min ∆(G0 ) ≤ |G0 | − 2 ≤ n − 4 ≤ n − 2 by Lemma
2.3(3).
4. Inverse results on ∆∗ (H). Let G be a finite abelian group. In this
section we study the structure of minimal non-half-factorial subsets G0 ⊂ G
with min ∆(G0 ) = max ∆∗ (G). These structural investigations were started
by Schmid who obtained a characterization in case exp(G) − 2 > m(G)
(Lemma 4.1(1)). Our main result in this section is Theorem 4.5. All examples
of minimal non-half-factorial subsets G0 ⊂ G with min ∆(G0 ) = max ∆∗ (G)
known so far are simple (in the sense of Remark 4.6), and it has been
conjectured that all such sets are simple. We provide the first example of a
set G0 which is not simple (Remark 4.6).
Lemma 4.1. Let G be a finite abelian group with |G| > 2, exp(G) = n,
r(G) = r, and let G0 ⊂ G be a subset with min ∆(G0 ) = max ∆∗ (G).
(1) Suppose that m(G) < n − 2. Then G0 is indecomposable if and only
if G0 = {g, −g} for some g ∈ G with ord(g) = n.
(2) Suppose that r ≤ n − 1. Then G0 is minimal non-half-factorial but
not an LCN-set if and only if G0 = {g, −g} for some g ∈ G with
ord(g) = n.
Proof. (1) See [30, Theorem 5.1].
(2) Since n = 2 implies r = 1 and |G| = 2, it follows that n ≥ 3. By Theorem 1.1, we have min ∆(G0 ) = n−2. Obviously, the set {−g, g}, with g ∈ G
and ord(g) = n, is a minimal non-half-factorial set with min ∆({−g, g}) =
n − 2 but not an LCN-set.
Conversely, let G0 be minimal non-half-factorial but not an LCN-set.
Then there exists A ∈ A(G0 ) with k(A) < 1. Since {n, nk(A)} ⊂ L(An ), it
follows that n−2 | n(k(A)−1), whence k(A) = 2/n. Consequently, A = (−g)g
for some g with ord(g) = n. Thus {−g, g} ⊂ G0 , and since G0 is minimal
non-half-factorial, equality follows.
Lemma 4.2. Let G be a finite abelian group with exp(G) = n and
r(G) = r.
16
A. Geroldinger and Q. Zhong
(1) Let G0 ⊂ G be a minimal non-half-factorial LCN-set with min ∆(G0 )
= max ∆∗ (G). Then |G0 | = r + 1, r ≥ n − 1, and for any distinct
h, h0 ∈ G0 we have h 6∈ hG0 \ {h, h0 }i.
(2) If r ≤ n − 2, then m(G) ≤ n − 3.
(3) If n ≥ 5 and r ≤ n − 3, then m(G) ≤ n − 4.
Proof. (1) We have min ∆(G0 ) ≤ |G0 | − 2 by Lemma 2.3(3), and moreover min ∆(G0 ) = max{n − 2, r − 1} by Theorem 1.1.
By Lemma 2.6(2) (properties (a) and (c)), we may assume that for each
g ∈ G0 we have g ∈ hG0 \ {g}i.
Case 1: There is a subset G2 ⊂ G0 such that hG2 i = hG0 i and |G2 | ≤
|G0 | − 2. The existence of G2 implies that G is isomorphic neither to C3 nor
to C2 ⊕C2 nor to C3 ⊕C3 (this is clear for the first two groups; to exclude the
case C3 ⊕ C3 , use again [27, Corollary 3.1] which says that k(U ) ∈ N for each
U ∈ A(G0 )). By Lemma 3.2, we know that min ∆(G0 ) ≤ max{n − 4, 1} <
max{n − 2, r − 1} = min ∆(G0 ), a contradiction.
Case 2: Every subset G1 ⊂ G0 with |G1 | = |G0 |−1 is a minimal generating set of hG0 i. Then for any distinct h, h0 ∈ G0 we have h 6∈ hG0 \ {h, h0 }i.
Assume to the contrary that |G0 | ≥ r + 2. Since r + 1 ≤ |G0 | − 1 ≤
∗
r (hG0 i), it follows by [13, Lemma A.6.2] that hG0 i is not a p-group. Since G0
is a minimal non-half-factorial subset, there exists an atom A with supp(A)
= G0 and hence G0 contains an element whose order is not a prime power.
Thus, by Lemma 3.1 we infer that |G0 | ≤ n − 2, and hence min ∆(G0 ) ≤
|G0 | − 2 ≤ n − 4, a contradiction.
Therefore |G0 | ≤ r+1. Then max{n−2, r−1} = min ∆(G0 ) ≤ |G0 |−2 ≤
r − 1, so we must have |G0 | = r + 1 and r ≥ n − 1.
(2) Assume to the contrary that r ≤ n − 2 and m(G) ≥ n − 2. Then
by Theorem 1.1, max ∆∗ (G) = max{r − 1, n − 2} = n − 2. Since m(G) ≥
n − 2, there is a minimal non-half-factorial LCN-set G0 with min ∆(G0 ) =
max ∆∗ (G), and then (1) implies that r ≥ n − 1, a contradiction.
(3) Let G0 ⊂ G be a non-half-factorial LCN-subset. We need to prove
that min ∆(G0 ) ≤ n − 4. Without restriction we may suppose that G0
is minimal non-half-factorial, which implies that G0 is indecomposable by
Lemma 2.5. By Lemma 2.6(2) we may suppose that for each g ∈ G0 we
have g ∈ hG0 \ {g}i. Suppose that the order of each element of G0 is a prime
power. Since G0 is indecomposable, Lemma 2.5 implies that each order is a
power of a fixed prime p ∈ P, and thus hG0 i is a p-group. By Proposition 2.4,
we infer that
min ∆(G0 ) ≤ m(hG0 i) = r(hG0 i) − 1 ≤ r(G) − 1 ≤ n − 4.
From now on we suppose that there is a g ∈ G0 whose order is not a prime
The set of minimal distances in Krull monoids
17
power. If |G0 | ≤ n − 2, then min ∆(G0 ) ≤ |G0 | − 2 ≤ n − 4 by Lemma 2.3(3)
Thus we may suppose that |G0 | ≥ n − 1 ≥ r + 2 and we distinguish two
cases.
Case 1: There exists a subset G2 ⊂ G0 such that hG2 i = hG0 i and
|G2 | ≤ |G0 | − 2. Then Lemma 3.2 implies that min ∆(G0 ) ≤ n − 4.
Case 2: Every subset G1 ⊂ G0 with |G1 | = |G0 | − 1 is a minimal generating set of hG0 i. Then for each h ∈ G0 , G0 \ {h} is half-factorial and
h ∈
/ hG0 \ {h, h0 }i for any h0 ∈ G0 \ {h}. Thus Lemma 3.1 implies that
|G0 | ≤ n − 2, a contradiction.
Lemma 4.3. Let G be a finite abelian group with exp(G) = n and
r(G) = r, and let G0 ⊂ G be a minimal non-half-factorial LCN-set with
min ∆(G0 ) = max ∆∗ (G).
(1) If A ∈ A(G0 ) with k(A) = 1, then |supp(A)| ≤ n/2.
(2) If A ∈ A(G0 ) with
> 1, then k(A) < r and SA−1 is also an
Q k(A) ord(g)
atom where S = g∈G0 g
.
Proof. By Lemma 4.2, we have r ≥ n − 1, |G0 | = r + 1, and for each
h ∈ G0 , h 6∈ hG0 \ {h, h0 }i for any h0 ∈ G0 \ {h}. Let A ∈ A(G0 ).
(1) Since k(A) = 1, it follows that |supp(A)| ≤ |A| ≤ n. Assume that
|supp(A)| = n. Then vg (A) = 1 for each g ∈ supp(A). Since G0 is a minimal non-half-factorial LCN-set, there is a V ∈ A(G0 ) with k(V ) > 1 and
supp(V ) = G0 . Therefore A | V , a contradiction.
Thus |supp(A)| ≤ n − 1, whence supp(A) ( G0 . Therefore Lemma
2.6.1 implies that gcd(vg (A), ord(g)) > 1 for each g ∈ supp(A), and hence
|supp(A)| ≤ |A|/2 ≤ n/2.
(2) Let A ∈ A(G0 ) with k(A) > 1. Then A | S, r + 1 = |G0 | = max L(S),
and L(S) \ {r + 1} =
6 ∅. By Theorem 1.1, we have min ∆(G0 ) = r − 1, hence
L(S) = {2, r + 1}, and thus SA−1 is an atom.
If k(SA−1 ) = 1, then (1) implies that |supp(SA−1 )| ≤ n/2, but on the
other hand |supp(SA−1 )| = |G0 | = r + 1 ≥ n, a contradiction.
Therefore k(SA−1 ) > 1 and hence r+1 = k(S) = k(A)+k(SA−1 ) implies
that k(A) < r.
Lemma 4.4. Let G be a finite abelian group with exp(G) = n and
r(G) = r, and let G0 ⊂ G be a minimal non-half-factorial LCN-set with
min ∆(G0 ) = max ∆∗ (G). Let g ∈ G0 with g ∈ hG0 \ {g}i, and let d ∈
[1, ord(g)] be minimal such that dg ∈ hE ∗ i, where the minimum is taken
over all subsets E ∗ ( G0 \ {g}. Then d | ord(g) and:
(1) Let k ∈ [1, ord(g) − 1] with d - k. Then there is an atom A with
vg (A) =Qk and k(A) > 1. If B ∈ B(G0 ) with vg (B) = k and B
divides g∈G0 g ord(g) , then B is an atom.
18
A. Geroldinger and Q. Zhong
(2) If A1 , A2 are atoms with vg (A1 ) ≡ vg (A2 ) mod d, then k(A1 ) =
k(A2 ).
Proof. Lemma 4.2 yields |G0 | = r + 1 and Q
r ≥ n − 1. The minimality of
d and (2.1) imply that d | ord(g). We set S = h∈G0 hord(h) .
(1) Since g ∈ hG0 \ {g}i, there is a zero-sum sequence B such that
vg (B) = k, and we choose B with minimal length |B|. Thus B | S, and it
remains to prove that B is an atom with k(B) > 1.
We set B = A1 · . . . · As with s ∈ N and atoms A1 , . . . , As . Then vg (A1 ) +
· · · + vg (As ) = vg (B) = k. Since d - k, there is an i ∈ [1, s] such that
d - vg (Ai ).
Assume to the contrary that k(Ai ) = 1. Then |supp(Ai )| ≤ n/2 by
Lemma 4.3(1). By the definition of d, there exists an atom A0 such that
vg (A0 ) = d and k(A0 ) = 1, which implies |supp(A0 )| ≤ n/2 by Lemma 4.3(1).
Hence gcd(d, vg (Ai )) < d, gcd(d, vg (Ai ))g ∈ hsupp(Ai ) ∪ supp(A0 ) \ {g}i, and
|supp(Ai ) ∪ supp(A0 ) \ {g}| ≤ n − 2 < r < |G0 |, contradicting the choice of d.
Thus k(Ai ) > 1. Since k ≤ ord(g) − 1, it follows that SB −1 6= 1. Since
−1
SAi = (SB −1 )(BA−1
i ) is an atom by Lemma 4.3(2), we infer that B = Ai
is an atom with k(B) > 1.
2. Let A1 ∈ A(G0 ). We assert that k(A1 ) = k(A2 ) for all A2 ∈ A(G0 )
with vg (A1 ) ≡ vg (A2 ) mod d. We distinguish two cases.
Case 1: d | vg (A1 ). There is an A ∈ A(G0 ) with vg (A) = d and k(A) = 1.
It is sufficient to show that k(A1 ) = 1. There are l ∈ N and V1 , . . . , Vl ∈
A(G0 \ {g}) (hence k(V1 ) = · · · = k(Vl ) = 1) such that
A1 A
ord(g)−vg (A1 )
d
= g ord(g) V1 · . . . · Vl ,
so k(A1 ) = 1 + l −
ord(g) − vg (A1 )
.
d
Furthermore, min ∆(G0 ) = r − 1 divides
ord(g) − vg (A1 )
(l + 1) − 1 +
= k(A1 ) − 1.
d
Since k(A1 ) < r by Lemma 4.3, it follows that k(A1 ) = 1.
Case 2: d - vg (A1 ). Let d0 ∈ [1, d − 1] be such that vg (A1 ) ≡ d0 mod d.
By (1), there are atoms Bl such that vg (Bl ) = d0 + ld for all l ∈ N0 with
d0 + ld < ord(g). Thus by an inductive argument it is sufficient to prove
the assertion for those atoms A2 with vg (A2 ) = vg (A1 ), and those with
vg (A2 ) = vg (A1 ) + d.
Suppose that vg (A1 ) = vg (A2 ). By (1), there is an atom V such that
vg (V ) = ord(g)−vg (A1 ) and k(V ) > 1. Then there are l ∈ N and V1 , . . . , Vl ∈
A(G0 \ {g}) such that A1 V = g ord(g) V1 · . . . · Vl , and hence k(A1 ) + k(V ) =
P
1+ li=1 k(Vi ) = l+1. Since k(V ) > 1, we have l > 1. Since min ∆(G0 ) = r−1
divides l − 1, either l = r or l ≥ 2r − 1. If l ≥ 2r − 1, then k(A1 ) ≥ r or
The set of minimal distances in Krull monoids
19
k(V ) ≥ r, contrary to Lemma 4.3. Therefore k(A1 ) + k(V ) = r + 1 =
k(A2 ) + k(V ), and hence k(A1 ) = k(A2 ).
Suppose that vg (A1 ) = vg (A2 ) + d. Let E ( G0 \ {g} be such that
dg ∈ hEi. Then there is an A ∈ A(E ∪ {g}) with vg (A) = d, and clearly
k(A) = 1. Let V1 , . . . , Vt be all the atoms with Vν | A2 A and |supp(Vν )| = 1
for all ν ∈ [1, t]. Since vg (A2 A) = vg (A1 ) < ord(g), it follows that B =
A2 A(V1 · . . . · Vt )−1 divides S and that vg (B) = vg (A1 ). Therefore (2) implies
that B is an atom, and by Step 1 we obtain k(B) = k(A1 ).
If t ≥ 2, then A2 A = BV1 · . . . · Vt implies t ≥ 1 + min ∆(G0 ) = r,
and thus k(A2 ) ≥ r, contradicting Lemma 4.3. Therefore t = 1, and thus
k(A2 ) + 1 = k(B) + 1 = k(A1 ) + 1.
Theorem 4.5. Let G be a finite abelian group with exp(G) = n and
r(G) = r, and let G0 ⊂ G be a minimal non-half-factorial set with min ∆(G0 )
= max ∆∗ (G).
(1) If r < n − 1, then there exists g ∈ G with ord(g) = n such that
G0 = {g, −g}.
(2) Let r = n − 1. If G0 is not an LCN-set, then there exists g ∈ G with
ord(g) = n such that G0 = {g, −g}. If G0 is an LCN-set, then |G0 | =
r + 1, and for any distinct h, h0 ∈ G0 we have h 6∈ hG0 \ {h, h0 }i.
(3) If r ≥ n, then G0 is an LCN-set with |G0 | = r + 1, and for any
distinct h, h0 ∈ G0 we have h 6∈ hG0 \ {h, h0 }i.
(4) If r ≥ n−1, G0 is an LCN-set, and n is odd, then there exists g ∈ G0
such that G0 \ {g} is independent.
Proof. (1) Suppose that r < n − 1. Then Lemma 4.2 implies that G0 is
not an LCN-set. Thus Lemma 4.1(2) shows that G0 has the asserted form.
(2) If G0 is not an LCN-set, then the assertion follows from Lemma
4.1(2); otherwise it follows from Lemma 4.2(1).
(3) Suppose that r ≥ n. Then Theorem 1.1 implies that min ∆(G0 ) =
max ∆∗ (G) = r − 1. Thus Lemma 2.3(3)(a) shows that G0 is an LCN-set.
Hence the assertion follows from Lemma 4.2(1).
(4) Let r ≥ n − 1, G0 be an LCN-set, and suppose that n is odd. By
Lemma 2.6(2) (properties (a) and (d)), we may suppose without restriction
that g ∈ hG0 \ {g}i for each g ∈ G0 . Lemma 4.2 implies that |G0 | = r + 1
and for each g ∈ G0 we have g 6∈ hEi for any E ( G0 \ {g}.
Assume to the contrary that G0 \{h} is dependent for each h ∈ G0 . Then
there exist g ∈ G0 , d ∈ [2, ord(g) − 1], and E ( G0 \ {g} such that dg ∈ hEi.
Now let d ∈ N be minimal over all configurations (g, E, d), and fix g, E
corresponding to d. It follows that we have an atom A with supp(A) ( G0
and vg (A) = d. By Lemma 4.4, d | ord(g), and hence d ≥ 3 because n is
odd.
20
A. Geroldinger and Q. Zhong
Since G0 \{g} is dependent, there exists U 0 ∈ A(G0 \{g}) with |supp(U 0 )|
> 1. Thus, by (2.1), there exist U ∈ A(G0 \ {g}) and h ∈ supp(U ) such that
vh (U ) ≤ ord(h)/2 and vh (U ) | ord(h).
By Lemma 4.4(1), there are atoms A1 , . . . , Ad−1 with vg (Ai ) = i and
k(Ai ) > 1 for each i ∈ [1, d−1], and we choose each Ai with vh (Ai ) minimal.
We prove the following assertion:
A. For each i ∈ [1, d − 1], we have vh (Ai ) < vh (U ) ≤ ord(h)/2.
Proof of A. Assume to the contrary that there is an i ∈ [1, d − 1] such
that vh (Ai ) ≥ vh (U ). Then
Y ord(h0 )
.
h∈
/ F = {h0 ∈ supp(U ) | vh0 (Ai ) < vh0 (U )} and U | Ai
h0
h0 ∈F
Q
0
Hence Ai h0 ∈F h0 ord(h ) = U Bi for some zero-sum sequence Bi . By Lemma 4.4
(items (1) and (2)), Bi is an atom with i = vg (Ai ) = vg (Bi ) and with
k(Bi ) = k(Ai ) > 1. Since vh (Ai ) > vh (Bi ), this contradicts the choice
of Ai . Proof of A
Let j ∈ [1, d − 1] be such that k(Aj ) = min{k(A1 ), . . . , k(Ad−1 )}.
Suppose that j ≥ 2. Let V1 , . . . , Vt be all the atoms with Vs | A1 Aj−1 and
|supp(Vs )| = 1 for all s ∈ [1, t]. Then B = A1 Aj−1 (V1 ·. . .·Vt )−1 is an atom by
Lemma 4.4(1). Since vg (A1 Aj−1 ) = j < ord(g), vh (A1 Aj−1 ) < ord(h), and
vf (A1 Aj−1 ) < 2 ord(f ) for all f ∈ G0 \ {g, h}, it follows that t ≤ |G0 | − 2 =
r − 1. Since min ∆(G0 ) = r − 1 and A1 Aj−1 = V1 · . . . · Vt B, we must have
t = 1. Therefore k(A1 ) + k(Aj−1 ) = 1 + k(B), whence k(B) > k(Aj−1 ). Since
vg (B) = vg (V1 B) = vg (A1 Aj−1 ) = j = vg (Aj ),
Lemma 4.4(2) implies that k(B) = k(Aj ) = min{k(A1 ), . . . , k(Ad−1 )}, a contradiction.
Suppose that j = 1. Let V1 , . . . , Vt be all the atoms with Vs | A2 Ad−1 and
|supp(Vs )| = 1 for all s ∈ [1, t]. Then B = A2 Ad−1 (V1 · . . . · Vt )−1 is an atom
by Lemma 4.4(1). Since vg (A2 Ad−1 ) = d+1 < ord(g), vh (A2 Ad−1 ) < ord(h),
and vf (A1 Aj−1 ) < 2 ord(f ) for all f ∈ G0 \{g, h}, it follows that t ≤ |G0 |−2
≤ r − 1. Since min ∆(G0 ) = r − 1 and A2 Ad−1 = V1 · . . . · Vt B, we must have
t = 1. Therefore k(A2 ) + k(Ad−1 ) = 1 + k(B), whence k(B) > k(A2 ). Since
vg (B) = vg (V1 B) = vg (A2 Ad−1 ) = d + 1 ≡ 1 = vg (A1 ) mod d,
Lemma 4.4(2) implies that k(B) = k(A1 ) = min{k(A1 ), . . . , k(Ad−1 )}, a
contradiction.
In the following remark we provide the first example of a minimal nonhalf-factorial subset G0 with min ∆(G0 ) = max ∆∗ (G) which is not simple.
Furthermore, we provide an example showing that the structural statement
The set of minimal distances in Krull monoids
21
given in Theorem 4.5(4) does not hold without the assumption that the
exponent is odd.
Remarks 4.6. Following Schmid, we say that a nonempty subset G0 ⊂
G \ {0} is simple if there exists g ∈ G0 such that G0 \ {g} is independent
and g ∈ hG0 \ {g}i, but g ∈
/ hEi for any subset E ( G0 \ {g}.
If G0 is a simple subset, then |G0 | ≤ r∗ (G)+1 and G0 is indecomposable.
Moreover, if G1 ⊂ G is such that any proper subset of G1 is independent,
then there is a subset G0 and a transfer homomorphism θ : B(G1 ) → B(G0 )
where G0 \ {0} is simple or independent (for all this see [26, Section 4]).
Furthermore, [26, Theorem 4.7] provides an intrinsic description of the sets
of atoms of a simple set.
In elementary p-groups, every minimal non-half-factorial subset is simple
[26, Lemma 4.4], and so far there are no examples of minimal non-halffactorial sets G0 with min ∆(G0 ) = max ∆∗ (G) which are not simple.
1. Let G = C9r−1 ⊕ C27 with r ≥ 26, and let (e1 , . . . , er ) be a basis of G
with ord(ei ) = 9 for i ∈ [1, r − 1] and ord(er ) = 27. Then max ∆∗ (G) = r − 1
by Theorem 1.1. We set
G0 = {3e1 , . . . , 3er−1 , er , g}
with g = e1 + · · · + er .
Then (er , g) is not independent, G0 \ {g} and G0 \ {er } are independent,
but g ∈
/ hG0 \ {g}i and er ∈
/ hG0 \ {er }i. Therefore G0 is not simple. It
remains to show that min ∆(G0 ) ≥ r − 1: then G0 is minimal non-halffactorial (because every proper subset is half-factorial) and min ∆(G0 ) =
r − 1 (because max ∆∗ (G) = r − 1).
We have
27 9 18 18 9
W1 = {A ∈ A(G0 ) | k(A) = 1} = {(3e1 )3 , . . . , (3er−1 )3 , e27
r , g , g er , g er },
W2 = {A ∈ A(G0 ) | k(A) > 1}
2
2
6 21
= {A3 = g 3 e24
r (3e1 ) · . . . · (3er−1 ) , A6 = g er (3e1 ) · . . . · (3er−1 ),
2
2
15 12
A12 = g 12 e15
r (3e1 ) · . . . · (3er−1 ) , A15 = g er (3e1 ) · . . . · (3er−1 ),
A21 = g 21 e6r (3e1 )2 · . . . · (3er−1 )2 , A24 = g 24 e3r (3e1 ) · . . . · (3er−1 )},
and k(A3 ) = k(A12 ) = k(A21 ) = (2r + 1)/3, k(A6 ) = k(A15 ) = k(A24 ) =
(r + 2)/3. For any d ∈ ∆(G0 ), there exists B ∈ B(G0 ) that has two such
factorizations, say
B = U1 · . . . · Us V1 · . . . · Vt W1 · . . . · Wu = X1 · . . . · Xs0 Y1 · . . . · Yt0 Z1 · . . . · Zu0
where all Ui , Vj , Wk , Xi0 , Yj 0 , Zk0 are atoms, s, t, u, s0 , t0 , u0 ∈ N0 with d =
(s + t + u) − (s0 + t0 + u0 ), k(U1 ) = · · · = k(Us ) = k(X1 ) = · · · = k(Xs0 ) =
(2r + 1)/3, k(V1 ) = · · · = k(Vt ) = k(Y1 ) = · · · = k(Yt0 ) = (r + 2)/2, and
22
A. Geroldinger and Q. Zhong
k(W1 ) = · · · = k(Wu ) = k(Z1 ) = · · · = k(Zu0 ) = 1. This implies that
2r + 1
r+2
2r + 1
r+2
k(B) = s
+t
+ u = s0
+ t0
+ u0
3
3
3
3
and v3e1 (B) ≡ 2s + t ≡ 2s0 + t0 mod 3. Since d = (s + t + u) − (s0 + t0 + u0 ) =
r−1
0
0
0
0
3 ((t − t) + 2(s − s)) > 0, we conclude that (t − t) + 2(s − s) ≥ 3 and
hence d ≥ r − 1.
2. We provide an example of a minimal non-half-factorial LCN-set G0
with min ∆(G0 ) = max ∆∗ (G) in a group G of even exponent which has
no element g ∈ G0 such that G0 \ {g} is independent. In particular, G0 is
not simple and the assumption in Theorem 4.5(4) that the exponent of the
group is odd cannot be cancelled.
Let G = C2r−2 ⊕ C4 ⊕ C4 with r ≥ 3, and let (e1 , . . . , er ) be a basis of G
with ord(ei ) = 2 for i ∈ [1, r − 2] and ord(er−1 ) = ord(er ) = 4. We set
G0 = {e1 , . . . , er−3 , er−2 + er−1 , er−1 , er , g},
g = e1 + · · · + er−2 + er .
Since (er−2 + er−1 , er−1 ) is dependent and (er , g) is dependent, there is no
h ∈ G0 such that G0 \ {h} is independent. We have
W1 = {A ∈ A(G0 ) | k(A) = 1}
= {e21 , . . . , e2r−3 , (er−2 + er−1 )4 , e4r−1 ,
e4r , g 4 , (er−2 + er−1 )2 e2r−1 , g 2 e2r },
W2 = {A ∈ A(G0 ) | k(A) > 1}
= {A1 = ge3r (er−2 + er−1 )e3r−1 e1 · . . . · er−3 ,
B1 = ge3r (er−2 + er−1 )3 er−1 e1 · . . . · er−3 ,
A3 = g 3 er (er−2 + er−1 )e3r−1 e1 · . . . · er−3 ,
B3 = g 3 er (er−2 + er−1 )3 er−1 e1 · . . . · er−3 },
and k(A1 ) = k(A3 ) = k(B1 ) = k(B3 ) = (r + 1)/2. Theorem 1.1 implies that
max ∆∗ (G) = r − 1, and thus it remains to show that min ∆(G0 ) = r − 1.
For any d ∈ ∆(G0 ), there exists B ∈ B(G0 ) with two such factorizations,
say
B = U1 · . . . · Us V1 · . . . · Vt = X1 · . . . · Xu Y1 · . . . · Yv
where all Ui , Vj , Xk , Yl are atoms, s, t, u, v ∈ N0 with d = u + v − (s + t),
k(U1 ) = · · · = k(Us ) = k(X1 ) = · · · = k(Xu ) = 1, and k(V1 ) = · · · = k(Vt ) =
k(Y1 ) = · · · = k(Yv ) = (r + 1)/2. This implies that
r+1
r+1
=u+v
k(B) = s + t
2
2
and vg (B) ≡ t ≡ v mod 2. Since d = (v + u) − (s + t) = (t − v) r−1
2 > 0, we
infer that t − v ≥ 2 and hence d ≥ r − 1.
The set of minimal distances in Krull monoids
23
Acknowledgements. The referee read the manuscript twice with utmost care. We would like to thank him/her for all the work which helped to
improve the clarity of the arguments and the presentation of the paper. This
work was supported by the Austrian Science Fund FWF, Project Number
M1641-N26.
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Alfred Geroldinger, Qinghai Zhong
Institute for Mathematics and Scientific Computing
University of Graz
NAWI Graz
Heinrichstrasse 36
8010 Graz, Austria
E-mail: [email protected]
[email protected]
Abstract (will appear on the journal’s web site only)
Let H be a Krull monoid with class group G. Then every nonunit a ∈ H
can be written as a finite product of atoms, say a = u1 · . . . · uk . The set
L(a) of all possible factorization lengths k is called the set of lengths of a. If
G is finite, then there is a constant M ∈ N such that all sets of lengths are
almost arithmetical multiprogressions with bound M and with difference
d ∈ ∆∗ (H), where ∆∗ (H) denotes the set of minimal distances of H. We
show that max ∆∗ (H) ≤ max{exp(G) − 2, r(G) − 1} and that equality holds
if every class of G contains a prime divisor, which holds true for holomorphy
rings in global fields.